The order of integration is nominal, I can change the projections if needed but I just chose z then y then x so that I have something to begin with.

In mos of the questions that I've done, the range of z has simply been an fairly simple and easy to see (eg. 0 <= z <= 8) but that isn't the case in this question. Since I can't find the range of z values I'll start with find the projection of the region onto the x-y plane.

I could see what the cylinders look like. It's really the region close to the intersection that I can't find a way to visualising. There's usually a way to combine the equations for the projections onto the 2D planes to find the intersection but I can't find one. Thanks for the help anyway.

In this particular case you don't need to combine the equations. Note that the first equation is independent of z. So to integrate over z first only the second equation will determine the upper and lower limits of z. Then, the second equation being independant of y, only the first equation will determine the upper and lower limits of y.